| L(s) = 1 | + (−2.81 + 1.04i)3-s + (4.33 + 2.48i)5-s − 10.8i·7-s + (6.83 − 5.85i)9-s − 15.2·11-s + 1.83·13-s + (−14.7 − 2.46i)15-s − 20.4·17-s + 13.6i·19-s + (11.3 + 30.6i)21-s − 19.4·23-s + (12.6 + 21.5i)25-s + (−13.1 + 23.6i)27-s + 24.5·29-s + 50.8·31-s + ⋯ |
| L(s) = 1 | + (−0.937 + 0.347i)3-s + (0.867 + 0.496i)5-s − 1.55i·7-s + (0.758 − 0.651i)9-s − 1.39·11-s + 0.141·13-s + (−0.986 − 0.164i)15-s − 1.20·17-s + 0.719i·19-s + (0.539 + 1.45i)21-s − 0.845·23-s + (0.506 + 0.862i)25-s + (−0.485 + 0.874i)27-s + 0.847·29-s + 1.64·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6152773150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6152773150\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.81 - 1.04i)T \) |
| 5 | \( 1 + (-4.33 - 2.48i)T \) |
| good | 7 | \( 1 + 10.8iT - 49T^{2} \) |
| 11 | \( 1 + 15.2T + 121T^{2} \) |
| 13 | \( 1 - 1.83T + 169T^{2} \) |
| 17 | \( 1 + 20.4T + 289T^{2} \) |
| 19 | \( 1 - 13.6iT - 361T^{2} \) |
| 23 | \( 1 + 19.4T + 529T^{2} \) |
| 29 | \( 1 - 24.5T + 841T^{2} \) |
| 31 | \( 1 - 50.8T + 961T^{2} \) |
| 37 | \( 1 + 16.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 26.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 63.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 31.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 81.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 50.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 135. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 81.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 75.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 124. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 58.1iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.12155819970751910942382567977, −9.894116156030188498716574459178, −8.381298859959055629313244397271, −7.38218812752718493997816337673, −6.60176299986867113211702399364, −5.91856596607004805885147489105, −4.83553695836252975053773912201, −4.10201605919153545678616282028, −2.74269584131233536019580645786, −1.21786399116802813880416903622,
0.22819892178849579095999726076, 1.95714873497583543484598172775, 2.61583227331853264445929106419, 4.75695115557340523565523072818, 5.16961462431151158782202465624, 6.07980831424165097053096535833, 6.60779492383943506593681154315, 8.023663679906532891454801690594, 8.663421812636730800447637979974, 9.613201958082926514953070977682
